The function f (x) is defined by the rule x2 +1 if x < 2 f (x) = if x = 2 %3D 4x – 3 if x > 2 - a) Calculate the left derivative and the right derivative of f at x = 2. You must use the limit definition because we did not establish rules for calculating one sided derivatives. b) Write a piecewise function for the derivative of f at all points in R.

If we need to calculate the derivative of a function at a point, there are two ways we can think about doing this. For example suppose f() = x – x, and we need to determine the value of f(4). One option is to just calculate the derivative at that point by plugging the point into the limit definition, like this: f(4 + h) – f(4) ( (4 + h)² – (4 + h)) - (4² – 4) f'(4) = lim (16 + 8h + h2 -4 – h) - (16 - 4) lim 7h + h2 = lim h(7+ h) lim h = lim h→0 h lim 7+h = 7. || h h h Another option is to calculate f'(x) in terms of x, and then plug in a value for x at the end, like this: f(x + h) – f(x) (2+ h) – (x + h)) - (2² – a) (22 + 2xh + h² – z – h) – (22 – 2) lim f'(x) = lim = lim 2xh + h2 - h h(2x + h - 1) = lim h 0 h h = lim = lim h h h that is to say, f'(x) = 2x – 1, and therefore f'(4) = 2(4) – 1 = 7. Notice that we get the answer f'(4) = 7 both ways. The second approach might look somewhat more complicated at first, but it turns out to be much more efficient if we ever need to know the…

Apply the limit definition to compute the derivative of each function. Use proper limit notation and show all details. Simplify your final answers as much as possible. a) g(x) = 4 + 3x b) h(x) = 5 - x c) p(x) = /5x? – 3 2.

Suppose there is a function called f(x). How do you find the slope, m, of the line that is tanget to function f(x) at x=2.